Learning from Liu Hui? a Different Way to Do Mathematics, Volume 49

Learning from Liu Hui? A Different Way to Do Mathematics Christopher Cullen ould we have done mathematics differently? But in fact it is not even necessary to imagine At a logical level this question is trivial: re- such an alternative history of mathematics, since Csearch mathematicians spend their time ex- one already exists. Ancient China developed its ploring all the ways one can “do it differently” and own mathematical culture based on a radically dif- then doing them. There are no signs that the math- ferent approach to the structuring of mathemati- ematical enterprise has any artificial barriers round cal thought. Unfortunately, not only western but it that stand in the way of this task. But my ques- also Chinese historians of mathematics have often tion refers to something a little less foundational. failed to see this point. As a result, early Chinese What if we make the “counterfactual move” of try- mathematicians have been portrayed as if they ing to imagine the history of mathematics with were doing the same job as Euclid, only—to be one of its great monuments no longer there—say blunt—a whole lot worse. In history, as in other dis- Euclid, for example. Could we imagine a possible ciplines, using the wrong tools for the job often history of mathematics without Euclid? By “math- breaks the thing you were trying to fix. But what ematics without Euclid” I do not of course mean were ancient Chinese mathematicians up to if they non-Euclidean geometry, but rather a mathematics were not playing the Euclidean game? I hope that stripped of the whole axiomatic-deductive scheme the answer to this question may be of more than for which Euclid’s writing served as the great ex- historical interest, since it bears directly on the emplar and entry point for generations of western pressing question of how mathematicians are to be mathematicians. At first glance the likely course of made and made more effectively. development of such a mathematics seems so dif- The Right Triangle Relation in China ferent from our own that it might deserve a place The problem of what Chinese mathematicians were in one of the more intellectually inclined episodes up to emerges very clearly if we start to ask about of Star Trek (Spock: “It’s mathematics, Jim, but the history of Pythagoras’ theorem in China. not as we know it”). Before we face the main issue, there are a few Christopher Cullen is senior lecturer in the History smaller problems in the way of our shift from west of Chinese Science and Medicine at the School of to east. For a start, we can hardly name the relation Oriental and African Studies, University of London, as used in China after a Greek thinker of around 520 England. His e-mail address is [email protected]. B.C. whose name was not even mentioned in China AUGUST 2002 NOTICES OF THE AMS 783 until many centuries later.1 More significantly, an- their present form by the first century A.D. In the cient Chinese mathematicians did not talk about Zhou bi the gougu relation is put to practical use right-angled triangles, because they did not talk in four instances,3 while in the Jiu zhang suan shu about triangles in any general sense: there is no an- it is the main theme in all the problems of the cient Chinese term corresponding to Greek 3 trigonon. They did, however, talk quite specifically See sections B11, B29 (twice), and B33 in the version about the ensemble of a horizontal ‘hook’ gou of Cullen (1996). Most of the Zhou bi is concerned with extended out at right angles from the foot of a ver- astronomy rather than mathematics per se. tical line, or ‘leg’ gu . Joining the ends of the hook and leg was what westerners call the hypotenuse, but the Chinese called the ‘bowstring’ xian . One does not, however, usually speak of gouguxian, but just of gougu ‘the hook and the leg’. When was the gougu relation (or whatever we choose to call it) first known and used in China? The first evidence that gou, gu, and xian were known to have a simple and useful relationship is found in the two earliest works of the classical Chinese mathematical canon, the Zhou bi (Gnomon of Zhou), and the Jiu zhang suan shu (Mathematical Methods in a Nine-fold Categorisation).2 Both these works were close to 1It is in any case well known that the “Pythagorean” relation between the sides of a right-angled triangle was used in Mesopotamia long before the study of mathematics began in the Hellenic world. See, for instance, Otto Neugebauer, The Exact Sciences in Antiquity (New York, 1969), pp. 34–6, and plate 6a showing a clay tablet of the Old Babylonian period (c. 1800–1600 B.C.) with the length of the diagonal Figure 1. of a unit square marked with a number equivalent to 1.414213 ... which is in error by one in only the seventh significant figure. As Neugebauer notes, Ptolemy used the same value in computing his table of chords two thousand years later. 2For a full translation and study of the first of these two works, see C. Cullen, Astronomy and Mathe- matics in Ancient China (Cambridge, 1996). The title is often found in the form Zhou bi suan jing (Mathematical canon of the Zhou gnomon), but the words suan jing were not added before approximately A.D. 600. The second work has been translated in Anthony W. C. Lun, J. N. Cross- ley, and Kangshen Shen, The Nine Chapters on the Mathematical Art: Companion and Commentary (Oxford, 1999). The title is translated by Lun et al. in what is now the traditional manner; the version given by me is intended to suggest more clearly the significance of the original Chinese. Figure 2. 784 NOTICES OF THE AMS VOLUME 49, NUMBER 7 ninth and final chapter. A manuscript of a pre- the last sections of this somewhat heterogeneous canonical mathematical text recently found in a work to be written and might have been added to tomb of the second century B.C. does not make use the book to make it more consonant with the cos- of the gougu relation, even where one might have mological numerology popular in the imperial court expected to find it, in problems relating to sawing in the early first century A.D.5 a square beam out of a round log.4 So it seems that However, putting the dating problem to one we have at least a rough fix on when gougu think- side, a further issue arises from the fact that the ing began in China. opening dialogue includes a passage which (though The situation has unfortunately been made much rather obscurely phrased) amounts to no more more confusing by the fact that the book tradi- than the statement of the gougu relation for the tionally placed first in the canonical mathematical case where the ‘hook’ is 3 units, the ‘leg’ is 4 units, series, the Zhou bi, was for centuries commonly and the ‘bowstring’ is 5 units. No premodern Chi- thought to date from the beginning of the Zhou dy- nese commentator has ever claimed to see anything nasty, around 1000 B.C., since it begins with a more substantial here, including some eighteenth- short dialogue between the Duke of Zhou (who and nineteenth-century scholars well versed in ruled as regent near the start of the Zhou dynasty) western as well as Chinese mathematics. Never- and Shang Gao, a sage of the preceding dynasty, theless, a number of twentieth-century historians in which the gougu relation is mentioned. No of mathematics in the East and West have felt scholar now believes in this early dating. I have ar- obliged to extract a “proof of Pythagoras” from this gued that this dialogue is in fact probably one of material by hook or by crook. I think the unfortu- nate results of this will be clear if I exhibit first my 4This text, bearing the title Suan shu shu own fairly literal translation of the relevant part of (A Book of Reckoning and Numbers) was recovered the dialogue, followed by one of the more creative in 1983 from a tomb at Zhangjiashan in versions that have been suggested:6 Hubei province, China. Evidence from other material found there suggests the tomb was closed #A1 [13b] Long ago, the Duke of Zhou in 186 B.C. Like most other Chinese books of its pe- asked Shang Gao , “I have riod, the Suan shu shu was written in ink on a se- heard, sir, that you excel in numbers. ries of bamboo strips bound together with strings, May I ask how Bao Xi 7 laid out the rather like a roller blind. The strings have perished successive degrees of the circumfer- over the centuries, leaving archaeologists with the ence of heaven in ancient times? Heaven tricky task of recovering the original order of 190 cannot be scaled like a staircase, and jumbled strips on which many characters were earth cannot be measured out with a partly or wholly obliterated. These and other diffi- footrule. Where do the numbers come culties no doubt explain why a preliminary tran- from?” scription of this material was published for the first time in 2000. The standard text is now that given #A2 [13f] Shang Gao replied, “The pat- in Peng Hao Zhangjiashan Han jian «Suan shu terns for these numbers come from the shu» zhu shi (The circle and the square.

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